Simultaneously numerically integrate coupled ODE's in MATLAB

In MATLAB, I need to numerically integrate two ODE's. Say I have variables time 't', x1(t) and x2(x1).

First differeential equation: contains dervative with respect to x1

Second differeential equation: contains dervative with respect to time 't'

time doesn't explicitly occur in either equation.

But since these are coupled equations, how do I simultaneously integrate at least numerically in MATLAB.

[John D'Errico]

So the ODEs are not ODEs. This is a system of PDEs.

You can use methods for the solution of PDEs. Entire books have been written on the subject. For example, approximate the derivatives using finite differences. If the system is linear, then the result will be a linear system of equations to then be solved.

[Ian DSouza]

Not quite. I can boil down the first ODE to "dx2/dx1 = some function of x1 and x2" and the second ODE to "dx1/dt = some other function of x1 and x2". But since x2 is a explicit function of x1 and x1 is an explicit function of t, so this is valid. I don't have to use partial derivatives because my dependent variables (x1 and x2) are explicit functions of at most one variable (with respect to which I am differentiating it). Ideally, this is what I'd like to do. Integrate ODE#1 to tell me how x2 varies with x1. Then integrate ODE#2 to tell me how x1 varies with respect to time. x1 vs. time is what I'm really after. Please let me know if you'd like to see the exact equations if this helps. I can update my question. Thanks.

Answer 1

[Teja Muppirala]

You can get dx2/dt by multiplying dx2/dx1 * dx1/dt.

As a simple example say (I'll use x and y instead of x1 and x2 cause it's easier to see):

dy/dx = x
dx/dt = t

Then the analytic solution (ignoring integration constants) is

x = t^2/2
y = x^2/2 = t^4/8

You can verify that dy/dt = t^3/2 = x*t = dy/dx * dx/dt.

So, the equations you'd put into the ODE solver would be:

dy/dt = x*t
dx/dt = x